Combining the results of Propositions REF  and REF , we know that for \(p_0\in M\) , either the function \(\alpha \)  as in (REF ) or the function \(\beta \)  as in (REF ), gives a smooth local coordinate system. Moreover these maps can be recovered fully from the data (REF ). As these two types of coordinate charts cover \(M\)  the smooth structure on \(M\)  is then the same as the maximal smooth atlas determined by these coordinate charts [1]}.
